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Mirrors > Home > NFE Home > Th. List > pssv | GIF version |
Description: Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.) |
Ref | Expression |
---|---|
pssv | ⊢ (A ⊊ V ↔ ¬ A = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3291 | . 2 ⊢ A ⊆ V | |
2 | dfpss2 3354 | . 2 ⊢ (A ⊊ V ↔ (A ⊆ V ∧ ¬ A = V)) | |
3 | 1, 2 | mpbiran 884 | 1 ⊢ (A ⊊ V ↔ ¬ A = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 176 = wceq 1642 Vcvv 2859 ⊆ wss 3257 ⊊ wpss 3258 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-pss 3261 |
This theorem is referenced by: (None) |
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